
It leaves our state of knowledge unchanged, reflecting complete ignorance about the time of nucleation. This leads us to impose an
(improper) uniform prior over proper (or e-folding) time, akin to a temporal Copernican principle.A uniform prior over the real line is of course ill-defined, so we
will need to introduce a late-time cutoff. In the limit that the cutoff is removed, the posterior probabilities will be well-defined and time-reparametrization invariant.
The uniform-time distribution is the least informative prior.

Alternatively, one might argue that a time translation increases the number of observers proportional to the growth in volume, hence \(\rho (t)\)  should grow exponentially in time. This effectively
corresponds to weighing occupational probabilities by physical volume. For \(\rho (t)\)  to be well-defined, it is necessary to impose a cutoff time \(t_{\rm c}\) . The resulting prior density is
exponentially peaked near \(t_c\) , to the extent that it can be well-approximated by a delta function.This embodies a version of the “youngness paradox" [1]}, though, as we will see, in our approach there is nothing paradoxical about this choice of prior. Thus the late-time or volume-weighted prior is maximally informative.

